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\begin{document}
\title{On Sinai-Bowen-Ruelle measures on horocycles
of 3-D Anosov flows}
\author{N.I. Chernov
\\ Department of Mathematics\\
University of Alabama at Birmingham\\
Birmingham, AL 35294
}
\date{\today}
\maketitle
\begin{abstract}
Let $\phi^t$ be a topologically mixing Anosov flow on a 3-D compact
manifolds $M$. Every unstable fiber (horocycle) of such a flow is
dense in $M$. Sinai proved in 1992 that the one-dimensional SBR
measures on long segments of unstable fibers converge uniformly
to the SBR measure of the flow. We establish an explicit bound
on the rate of convergence in terms of integrals of H\"older
continuous functions on $M$.
\end{abstract}
\centerline{\em Keywords: Anosov flows, horocycles, Sinai-Bowen-Ruelle
mwasures}
\centerline{\em AMS subject classification: 58F17}
%\newtheorem{theorem}{Theorem}[section]
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
%\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\section{Introduction}
Let $\phi^t:M\to M$ be a $C^2$ Anosov flow on a smooth compact 3-D
Riemannian manifold $M$. This means that $\phi^t$ has no
fixed points, and at every $x\in M$ there is a $D\phi^t$-invariant
splitting of the tangent space
\be
{\cal T}_x M=E_x^s\oplus E^u_x\oplus E^{\phi}_x
\label{EEE}
\ee
into stable, unstable and neutral (parallel to the direction of the
flow) one-dimensional subspaces. We assume that $\phi^t$ is
topologically mixing.
Let $\mu$ be the Sinai-Bowen-Ruelle (SBR) measure for $\phi^t$.
For topologically mixing flows, $\mu$ is the only invariant measure
whose conditional distributions on unstable fibers are absolutely
continuous with respect to the Riemannian length \cite{Si72}.
Also \cite{BR}, the SBR
measure is the weak limit of the measure $\phi^t\mu_0$, as $t\to
\infty$, for any smooth measure $\mu_0$ on $M$.
The SBR measure $\mu$ is a Gibbs measure \cite{Si72} and
Bernoulli \cite{Ra74}.
The measure $\mu$ can be approximated by its conditional
measures on one-dimensional unstable fibers as follows
\cite{Si92}. Let $x\in M$ and $R>0$. Denote by $W^u_{x,R}$
the segment of the unstable fiber through $x$
of length $R$ on which $x$ is the central point (equidistant from the
endpoints). Now, for any $t>0$ we take the normalized Lebesgue
measure on the curve $\phi^{-t}W^u_{x,R}$ and pull this measure back
onto $W^u_{x,R}$ under the action of $\phi^t$. We get a probability
measure on $W^u_{x,R}$ denoted by $\nu^u_{x,R,t}$. The weak limit
\be
\nu^u_{x,R}=\lim_{t\to\infty} \nu^u_{x,R,t}
\label{nuxR}
\ee
exists and is a smooth probability measure on $W^u_{x,R}$. It is
the SBR measure $\mu$ conditioned on the curve $W^u_{x,R}$.
The measures $\nu^u_{x,R}$ are invariant under the flow
$\phi^t$ in the following sense: if $\phi^t W^u_{x,R}\subset
W^u_{y,S}$ for some $y\in M$ and $S>0$, then the measure $\nu^u_{y,S}$
conditioned on $\phi^t W^u_{x,R}$ and pulled back under $\phi^{-t}$
will coincide with $\nu^u_{x,R}$. In particular, if $W^u_{x,R}
\subset W^u_{y,S}$, then $\nu^u_{y,S}$ conditioned on $W^u_{x,R}$
coincides with $\nu^u_{x,R}$. The density $f_{x,R}(y)$, $y\in W^u_{x,R}$
of the measure $\nu^u_{x,R}$ with respect to the Riemannian length
satisfies the Anosov-Sinai formula \cite{AS,Si92,Ch95}
\be
\frac{f_{x,R}(y_1)}{f_{x,R}(y_2)}
=\lim_{t\to -\infty}\frac{\Lambda_t^u(y_1)}{\Lambda_t^u(y_2)}
\label{ff}
\ee
where $\Lambda^u_t(y)$ is the Jacobian of the linear map $D\phi^t:
E^u_y\to E^u_{\phi^ty}$.
The topological mixing of $\phi^t$ means by definition that every
global unstable fiber
$\Gamma^u_x=\cup_RW^u_{x,R}$ is dense in $M$.
The K-mixing property of $\mu$ implies \cite{Si92} that the weak
limit of $\nu^u_{x,R}$, as $R\to\infty$, exists and coincides with
the SBR measure $\mu$ for a.e. point $x$. Sinai has proved \cite{Si92}
that the weak convergence $\nu^u_{x,R}\to\mu$, as $R\to\infty$,
occurs for {\em every} point $x\in M$ and is
uniform in $x$ in the following sense.
\begin{theorem}[Sinai, \cite{Si92}]
For any continuous function
$F(x)$ on $M$ and any $\varepsilon >0$ there is an $R_\varepsilon>0$
such that
\be
\left | \int_{W^u_{x,R}} F\, d\nu^u_{x,R} - \int_M F\, d\mu \right |
\leq \varepsilon
\label{Sin}
\ee
for all $x\in M$ and $R>R_\varepsilon$.
\end{theorem}
Technically, Sinai proved (\ref{Sin}) for geodesic flows on
surfaces of negative curvature, but his proof works for the
all Anosov flows discussed here without any changes.
Sinai termed the property (\ref{Sin}) the {\em uniform distribution}
of horocycles (this is the name for unstable and stable fibers
of geodesic flows). As Sinai pointed out \cite{Si92}, for geodesic flows
on noncompact surfaces of finite area the property (\ref{Sin}) fails --
every such surface has a finite number of families of closed (!)
horocycles. Nonclosed horocycles are still dense and uniformly
distributed \cite{Si92}. Sinai's theorem was motivated
by a remark made by Zagier \cite{Zag}, according to which
the Riemannian hypothesis for zeroes of $\zeta$-function could be
derived from the properties of closed horocycles on the modular surface.
We are going to estimate the rate of the weak convergence
$\nu^u_{x,R}\to \mu$. Our main result is the following.
\begin{theorem}
Let $\phi^t:M\to M$ be a topologically mixing $C^2$ Anosov flow
on a 3-D manifold, under the UNF assumption stated below. Let
$F$ be a H\"{o}lder continuous function with H\"older exponent
$\alpha>0$ on $M$. Then for all $x\in M$ and $R>1$ we have
\be
\left | \int_{W^u_{x,R}} F\, d\nu^u_{x,R} - \int_M F\, d\mu \right |
\leq C_{\phi,F}\cdot \exp\left [-\alpha d_\phi (\ln R)^{1/2}\right ]
\label{main}
\ee
where the factor $d_{\phi}$ depends on the flow $\phi^t$ alone,
and $C_{\phi,F}$ depends on both $\phi^t$ and $F$, but both
factors are independent of $R$ and $x$.
\label{tmmain}
\end{theorem}
We now formulate the assumption we call UNF - uniform nonintegrability
of foliations. We denote by $W^{u,s}_x$ the (one-dimensional)
local stable and unstable fibers through $x\in M$. We denote by
$$
W^{wu,ws}_x=\phi^{[-\varepsilon,\varepsilon]}W^{u,s}_x
$$
the weak local unstable and stable (two-dimensional) leaves
through $x$.
Let $U\subset M$ be an open set, small enough so that both families
of local stable and unstable fibers in $U$ are orientable. Fix some
orientations of those families in $U$. Let $y\in U$ and $\delta>0$
a small number. On the local unstable and stable fibers
$W_y^{u}$ and $W_y^{s}$ we take two positively oriented segments
of length $\delta$ starting at $y$ and terminating at some points
$y_1\in W_y^{u}$ and $y_2\in W_y^{s}$, respectively. It is clear
that the two points $y'=W^s_{y_1}\cap W^{wu}_{y_2}$ and
$y''=W^u_{y_2}\cap W^{ws}_{y_1}$ lie on the same orbit of
the flow, i.e. $y'=\phi^{\tau}y''$ for some small number $\tau=
\tau_y(\delta)$. We call this $\tau$ the {\it temporal distance}
between the local fibers $W^s_{y_1}$ and $W^u_{y_2}$, see also \cite{Ch95}.
The foliations by local stable and unstable fibers are said to
be {\it jointly integrable} \cite{Pl} in $U$ if $\tau_y(\delta)=0$
for all $y\in U$ and small $\delta>0$. In that case those
are subfoliations of the same $C^1$ foliation of $U$ by surfaces.
Plante's \cite{Pl} results imply that the flow $\phi^t$ is topologically
mixing iff there is an open domain $U\subset M$ where the stable
and unstable foliations are {\em not} jointly integrable. Motivated
by this, we call the next assumption the {\em uniform nonitegrability}
of stable and unstable foliations.\medskip
\noindent{\bf Assumption UNF.}
There are $\delta_0>0$ and an open domain $U\subset M$
where both families of stable and unstable fibers are
orientable, and for some orientation we have, at every
$y\in U$ and all $0<\delta<\delta_0$,
\be
0<\underline{d} < \frac{\tau_y(\delta)}{\delta^2}
< \overline{d} < \infty
\label{UNC}
\ee
where $\underline{d}$ and $\overline{d}$ do not depend on $y$. \medskip
This assumption was first introduced in \cite{Ch95}. Based on it, a
stretched exponential bound on correlation functions for the flow
$\phi^t$ was established.
It was also shown there that this assumption is always satisfied
for 3-D contact Anosov flows and, in particular, for all geodesic
flows on compact surfaces of (constant or variable) negative
curvature.
\begin{corollary}
Let $\phi^t:M\to M$ be a 3-D contact Anosov flow. Then the statement
of Theorem~\ref{tmmain} holds true. In particular, it holds for
geodesic flows on compact surfaces of negative curvature.
\end{corollary}
\section{Markov partitions into boxes}
Our proof of Theorem~\ref{tmmain} is based on Markov approximations
to Anosov flows developed in \cite{Ch95}. We will often refer to that
paper, and for the reader's convenience we use here the same notations.
One can thus consider this paper as a continuation of \cite{Ch95}.
The main construction of the paper \cite{Ch95} is an increasing sequence
of partitions of the manifold $M$ into small boxes, which
enjoy special Markov properties. We describe those partitions
below, in a slightly modified form adjusted to our current needs.
First, let ${\cal R}=\{R_1,\ldots,R_I\}$ be a Markov family
of rectangles in $M$ defined by Bowen \cite{Bo1}. Every
rectangle is a small $C^2$ compact surface in $M$ transversal to
the flow $\phi^t$. The boundary of each rectangle consists
of four curves, two lying on local unstable leaves of
$\phi^t$ and two others on local stable leaves.
Rectangles need not be foliated by stable or unstable fibers.
However, local stable and unstable leaves intersect a
rectangle in smooth curves that we call {\em induced} stable
and unstable fibers on the rectangle.
Let $\Omega=\cup R_i$. The surface $\Omega$ is a cross-section
for the flow $\phi^t$, see \cite{Bo1}. Denote by $T:\Omega\to\Omega$ the first
return map and by $l(x)>0$, $x\in\Omega$, the first return time.
The flow $\phi^t$ is then isomorphic to a suspension flow
built over the map $T:\Omega\to\Omega$ under the ceiling
function $l(x)$.
The function $l(x)$ is piecewise $C^2$ smooth with discontinuities
along a finite collection of induced stable fibers in the
rectangles of $\cal R$.
The function $l(x)$ on $\Omega$ is bounded away from $0$ and
$\infty$. The map $T:\Omega\to\Omega$ is piecewise $C^2$ smooth
and hyperbolic. The above mentioned induced fibers on
rectangles are just stable and unstable fibers for $T$.
The partition of $\Omega$ into rectangles $R_i\in\cal R$
is a Markov partition for $T$.
We fix the Markov family $\cal R$, assuming
its rectangles be small enough, see \cite{Bo1,Ch95}. In what follows,
we denote by $a_i,c_i,d_i$, for $i=1,2,\ldots$ and $z,t_0$
various positive constants determined only by the flow $\phi^t$
and the family $\cal R$.
Now let $m\geq 0$ be an integer-valued parameter. A refinement
${\cal A}_m$ of the Markov partition $\cal R$
was constructed in \cite{Ch95}, with the following properties.
First, ${\cal A}_m$ is a Markov partition itself,
its atoms are small subrectangles
in the rectangles $R_i\in\cal R$. For every rectangle $A\in{\cal A}_m$
we can find two integers $n_+(A)>0$ and $n_-(A)>0$ such that the
images $T^{n_+(A)}A$ and $T^{-n_-(A)}A$ are subrectangles
in some rectangles $R\in\cal R$, stretching
across those rectangles completely (from one boundary
curve to the opposite one). One can call the numbers $n_{\pm}(A)$
the {\em ranks} of the rectangle $A\in{\cal A}_m$.
In the constructions made in \cite{Ch95}, the ranks
$n_{\pm}(A)$ depended on $A\in{\cal A}_m$, but they were bounded
as follows:
$$
d_1m\leq n_{\pm}(A)\leq d_2m
$$
with some constants
$00$ we have $\hat{\phi}_m^tx
=\phi^{t'}x$ for some $t'\geq t$. It was shown in \cite{Ch95} that
\be
|t'-t|\leq c_2t e^{-a_2m}
\label{tt}
\ee
so that the points $\hat{\phi}_m^tx$ and $\phi^tx$ are
close for relatively small $t$.
Obviously, the flow $\hat{\phi}^t_m$ preserves the measure
$\hat{\mu}_m$. It was shown in \cite{Ch95} that $\hat{\phi}_m^t$ is
a hyperbolic flow with singularities. Its stable and unstable
fibers are time-shifts of stable and unstable fibers of
the map $T$, which we termed induced fibers.
The dynamics of the flow $\hat{\phi}^t_m$ has a clearly pronounced
``discrete'' or ``quantum'' character. Let $\hat{T}_m=\hat{\phi}_m^{\delta}$
(one can think of $\delta$ as a quantum of time). The map
$\hat{T}_m$ moves every box forward onto the next one in
$\hat{\cal A}_m$, except for the boxes whose top faces are
on the border of a component of $\hat{M}_m$.
Those border boxes are moved by $\hat{T}_m$ across the gaps and
their images will fill the boxes $X\in\hat{\cal A}_m$
adjacent to the surface $\Omega$ on the other side of the
gap. Due to these properties of
the dynamics $\hat{\phi}^t_m$ it can be well approximated
by a Markov chain, as it was shown in \cite{Ch95}.
We now summarize some of the results of \cite{Ch95}. Take an
arbitrary box $X_0\in\hat{\cal A}_m$, and let $t>0$. Consider
the conditional distribution $\hat{\mu}_m(\cdot /\hat{\phi}^t_mX_0)$ on
the boxes $X\in\hat{\cal A}_m$ defined by
$$
\hat{\mu}_m(X/\hat{\phi}^t_mX_0)=
\hat{\mu}_m(X\cap\hat{\phi}^t_mX_0)\cdot [\hat{\mu}_m(X_0)]^{-1}
$$
For a function $F$ on $M$, we define the average of $F$ with
respect to the above distribution by
$$
\langle F/\hat{\phi}_m^tX_0\rangle = \sum_{X\in\hat{\cal A}_m}
\bar{F}_m(X)\cdot \hat{\mu}_m(X/\hat{\phi}^t_mX_0)
$$
where
$$
\bar{F}_m(X)=[\hat{\mu}_m(X)]^{-1}\int_X F(x)\, d\hat{\mu}_m(x)
$$
\begin{theorem}
There are constants $z>0$ and $t_0>0$ such that for every
$t>t_0$ one can take $m=[z\sqrt{t}]$ and then for any
H\"older continuous function $F$ on $M$ one has
\be
\left | \int_{\hat{M}_m} F(x)\, d\hat{\mu}_m(x)-
\langle F/\hat{\phi}^t_mX\rangle \right |
\leq C_{F,1}\cdot c_3\cdot \exp(-\alpha a_3m)
\label{Xt}
\ee
for every box $X\in\hat{\cal A}_m$. Here $\alpha$ is the H\"older
exponent of $F$, and the factor $C_{F,1}>0$ depends on the
function $F$ alone.
\label{thuni}
\end{theorem}
This theorem actually says that the image $\hat{\phi}^t_mX$
is pretty much uniformly distributed over the space $\hat{M}_m$.
In other words, the conditional measure on $\hat{\phi}^t_mX$ reproduces
the invariant measure $\hat{\mu}_m$ of the flow $\hat{\phi}^t_m$
very accurately, as specified by (\ref{Xt}). We emphasize the
important relation $m=[z\sqrt{t}]$ between $m$ and $t$.
This theorem follows immediately from the results of \cite{Ch95},
but we should give a warning. The bounds on correlation functions
developed in \cite{Ch95} required the above `uniformity' of
the distribution of $\hat{\phi}^t_mX$ {\em on the average}
over the boxes $X\in\hat{\cal A}_m$, rather than for every
single box $X$. So, all the main estimates in \cite{Ch95}
are given by averaging over those boxes. Fortunately, the paper
\cite{Ch95} also contains a proof of the uniformity of the
distribution of $\hat{\phi}^t_mX$ for {\em every} box $X\in\hat{\cal A}_m$,
see remarks in the end of Section~16 and Theorem~6.1 in \cite{Ch95}.
By using the smallness of $\mu(M\setminus\hat{M}_m)$ and (\ref{tt})
we get
\begin{corollary}
Under the conditions of Theorem~\ref{thuni} we have
\be
\left | \int_{M} F(x)\, d\mu(x)-
(\mu(X))^{-1}\int_{\phi^tX} F(x)\, d\mu(x) \right |
\leq C_{F,2}\cdot c_4\cdot \exp(-\alpha a_4m)
\label{Xt1}
\ee
for every box $X\in\hat{\cal A}_m$. Here $\alpha$ is the H\"older
exponent of $F$, and the factor $C_{F,2}>0$ depends on the
function $F$ alone.
\label{cr1}
\end{corollary}
Moreover, we can modify the box $X$ here without harming the property
(\ref{Xt1}) so that the new box will be foliated by unstable fibers
in the following way. Let $x\in X$ and let $W^s_x(X)$ be
the smallest segment of the local
stable fiber through $x$ that terminates on the local unstable
leaves bounding the box $X$ (of course, $W^s_x(X)$ may go out of
the box $X$ and then terminate on the continuation of the
unstable leaf which contains a face
of $\partial X$). For every point $y\in W^s_x(X)$
we take a segment of the local unstable fiber $W^u_y(X)$ that,
in the same way as before, terminates on two local stable leaves
bounding $X$ (or their continuation beyond $\partial X$). The surface
$$
B_x^u(X)=\cup_{y\in W^s_x(X)} W^u_y(X)
$$
is foliated by unstable fibers. Sinai \cite{Si92} called such
surfaces $u$-cells. We now take $Y_x=\phi^{[0,\delta]}
B_x^u(X)$. This is a domain, bounded by two surfaces $B_x^u(X)$,
$\phi^\delta B_x^u(X)$,
two local stable leaves, and two local unstable leaves.
So we can call $Y_x$ a box. This is our modification of the
box $X$. The box $Y_x$ is obviously
foliated by unstable fibers of the flow $\phi^t$.
There is a natural one-to-one smooth correspondence $S:X\to Y_x$ which
preserves the measure, leaves every point $x\in X$ on its
trajectory and moves points no farther than by $c_5e^{-a_5m}$.
The map $S$ sends the bottom of the box $X$ onto the surface
$B_x^u(X)$. We then immediately obtain
\begin{corollary}
Under the conditions of Theorem~\ref{thuni} we have
\be
\left | \int_{M} F(x)\, d\mu(x)-
(\mu(Y_x))^{-1}\int_{\phi^tY_x} F(x)\, d\mu(x) \right |
\leq C_{F,3}c_6\cdot \exp(-\alpha a_6m)
\label{Xt2}
\ee
for the modification $Y_x$ of every box $X\in\hat{\cal A}_m$.
Here $\alpha$ is the H\"older exponent of $F$, and the
factor $C_{F,3}>0$ depends on the function $F$ alone.
\label{cr2}
\end{corollary}
\section{Proof of the main theorem}
The proof of Theorem~\ref{tmmain} is based on Corollary~\ref{cr2}
and a few relatively simple arguments.
First, we recall the notion of the holonomy map. For any
two close unstable fibers $W^u_1,W^u_2\in M$ the map
$H:W^u_1\to W^u_2$ defined by $H(y)=W^{ws}_y\cap W^u_2$ is
called canonical isomorphism, or holonomy map \cite{AS,Ch95}.
Its Jacobian, $DH$, with respect to the Riemannian length on
the curves $W^u_1,W^u_2$ is bounded away from $0$ and $\infty$.
Moreover, it is close to one if the fibers are close enough
to each other \cite{Ch95}. This property is commonly known
as the absolute continuity of stable and unstable foliations
\cite{AS,Ch95}.
The following lemma gives a specific bound on the Jacobian $DH$:
\begin{lemma}
There are constants $a,c>0$, determined by the flow $\phi^t$, such that
\be
\exp(-c\varepsilon^a)\leq DH \leq \exp(c\varepsilon^a)
\label{DHl}
\ee
where $\varepsilon=\mbox{\rm dist}(y,H(y))$.
\label{lmDh}
\end{lemma}
{\it Proof}. Put $y_\ast=W_y^s\cap W_{H(y)}^{wu}$.
Denote by $H_\ast$ the holonomy map $W_1^u\to W_{y_\ast}^u$.
There is a small $\tau_\ast$ such that $\phi^{\tau_\ast}
W_{y_\ast}^u=W_2^u$. Then we have $DH(y)=DH_\ast(y)\cdot
\Lambda^u_{\tau_\ast}(y_\ast)$. For the Jacobian $DH_\ast(y)$
an analog of Anosov-Sinai formula \cite{AS,Ch95} holds,
which says that $DH_\ast(y)=\lim_{t\to\infty} \Lambda^u_t(y)
/\Lambda^u_t(y_\ast)$. The existence of this limit and
its closeness to one required by (\ref{DHl})
follows from the facts that $y_\ast\in
W_y^s$ and the function $\Lambda_t^u(\cdot)$ is H\"older
continuous on $M$ for any $t$, see \cite{Ch95}. Lemma~\ref{lmDh}
is proved. \medskip
We now turn to the proof of Theorem~\ref{tmmain}.
Let $t>t_0$ and $m=[z\sqrt{t}]$, as in Corollary~\ref{cr2}.
Take an arbitrary box $X\in\hat{\cal A}_m$.
For any $x\in X$ the modified box $Y_x$ is foliated
by unstable fibers. Its image
$Y_{x,t}=\phi^tY_x$ is also a domain foliated by
unstable fibers. We denote the partition of
$Y_{x,t}$ into unstable fibers by $\xi^u(Y_{x,t})$.
Since $t\sim m^2$, the length of unstable fibers
$W^u\in\xi^u(Y_{x,t})$ grows exponentially in $m^2$:
\be
c_7e^{a_7m^2}\leq\, {\rm length}(W^u)\leq c_8e^{a_8m^2}
\label{leng}
\ee
It is easy to see that all the unstable fibers in the
partition $\xi^u(Y_{x,t})$ are canonically isomorphic. For
any two fibers $W^u_1,W^u_2\in\xi^u(Y_{x,t})$ and any point
$y\in W^u_1$ we have
$$
{\rm dist}(y,H(y))\leq c_9e^{-a_9m}
$$
Therefore, due to Lemma~\ref{lmDh},
\be
\exp(-c_{10}e^{-a_{10}m})\leq DH \leq \exp(c_{10}e^{-a_{10}m})
\label{DH}
\ee
We now condition the measure $\mu$ on the domain $Y_{x,t}$
with respect to the partition $\xi^u(Y_{x,t})$. It induces smooth
probability measures on the fibers $W^u\in\xi^u(Y_{x,t})$, which
we denote by $\nu_{W^u}^u$. The density $f_{W^u}(y)$, $y\in W^u$,
of the measure $\nu^u_{W^u}$ with respect to the Riemannian
length satisfies the ratio formula (\ref{ff}):
\be
\frac{f_{W^u}(y_1)}{f_{W^u}(y_2)} = \kappa_{W^u}(y_1,y_2) =
\lim_{\tau\to -\infty}\frac{\Lambda_\tau^u(y_1)}{\Lambda_\tau^u(y_2)}
\label{ff1}
\ee
The function $\kappa$ satisfies the rule $\kappa_{W^u}(y_1,y_2)
\cdot\kappa_{W^u}(y_2,y_3)=\kappa_{W^u}(y_1,y_3)$. The density
$f_{W^u}$ can be computed by
\be
f_{W^u}(y)=\frac{\kappa_{W^u}(y,y_0)}{\int_{W^u}
\kappa_{W^u}(y,y_0)\, dy}
\label{fint}
\ee
where $y_0\in W^u$ is any point, and the integration is performed
with respect to the Riemannian length on $W^u$.
Let $W^u_1,W^u_2\in\xi(Y_x)$ be two unstable fibers, and
$y_1,y_2\in W^u_1$. We will show that
\be
\frac{\kappa_{W_1^u}(y_1,y_2)}{\kappa_{W_2^u}(H(y_1),H(y_2))}
=e^{\varepsilon}
\label{kappa}
\ee
for some $|\varepsilon|\leq c_{11}e^{-a_{11}m}$. Indeed,
\be
\frac{\kappa_{W_1^u}(y_1,y_2)}{\kappa_{W_2^u}(H(y_1),H(y_2))}
=\frac{\Lambda_{-t}^u(y_1)}{\Lambda_{-t}^u(H(y_1))}
\frac{\Lambda_{-t}^u(H(y_2))}{\Lambda_{-t}^u(y_2)}\cdot
e^{\varepsilon'}
\label{LLLL}
\ee
for some $|\varepsilon|\leq c_{11}e^{-a_{11}m}$, because all four
points $\phi^{-t}y_i,\phi^{-t}H(y_i)$, $i=1,2$, lie in the small
box $Y_x$ whose size decrease exponentially in $m$. To estimate
the two fractions on the right-hand side of (\ref{LLLL}), it is
enough to note that the points $\phi^{-t}y_1,\phi^{-t}H(y_1)$
belong in one stable leaf in $Y_x$, as well as the points
$\phi^{-t}y_2,\phi^{-t}H(y_2)$, and use again the H\"older
continuity of the function $\Lambda_s^u(\cdot)$ for any $s$.
So, we get (\ref{kappa}).
Combining (\ref{DH}), (\ref{fint}) and (\ref{kappa}) gives the bound
\be
\exp(-c_{12}e^{-a_{12}m})\leq D_\ast H \leq \exp(c_{12}e^{-a_{12}m})
\label{DH2}
\ee
where
$$
D_\ast H(y)=\frac{f_{W_2^u}(H(y))}{f_{W_1^u}(y)}\cdot DH
$$
is the Jacobian of the holonomy map now measured
with respect to the induced measures $\nu^u_{W_1^u},\nu^u_{W^u_2}$
on the unstable fibers $W^u_1,W^u_2\in\xi^u(Y_{x,t})$,
rather than their Riemannian length.
Corollary~\ref{cr2} can be now reformulated:
\begin{corollary}
Under the conditions of Theorem~\ref{thuni}, for any unstable
fiber $W^u\in\xi^u(Y_{x,t})$ we have
\be
\left | \int_{M} F(x)\, d\mu(x)-
\int_{W^u} F\, d\nu^u_{W^u} \right |
\leq C_{F,4}\cdot c_{13}\cdot \exp(-\alpha a_{13}m)
\label{Xt3}
\ee
Here $\alpha$ is the H\"older exponent of $F$, and the
factor $C_{F,4}>0$ depends on the function $F$ alone.
\label{cr3}
\end{corollary}
This corollary, along with the bounds (\ref{leng}) complete
the proof of Theorem~\ref{tmmain} provided the unstable fiber
$W^u_{x,R}$ entering that theorem belongs in $\xi^u(Y_{x,t})$
for some box $X\in{\cal A}_m$ for some $m\geq 1$.
For a generic unstable fiber $W^u_{x,R}$ we first find a $t>0$
such that the preimage $\phi^{-t}W^u_{x,R}$ has a length of order
one (independently of $R$) and is located some positive distance
apart from the rectangles of the Markov family $\cal R$. Obviously,
$$
d_3\ln R\leq t\leq d_4\ln R
$$
Then we take $m=[z\sqrt{t}]$, and
the corresponding partition $\hat{\cal A}_m$ of $\hat{M}_m$ into boxes.
The stable leaves bounding those boxes will partition the curve
$\phi^{-t}W^u_{x,R}$ into short unstable fibers. Every short fiber,
except for the two on both ends of $\phi^{-t}W^u_{x,R}$, will
belong in some modified box defined in the end of the previous
section. Its image under $\phi^{t}$ will then satisfy the
bound (\ref{Xt3}). These images constitute nonoverlapping parts
of the given fiber $W^u_{x,R}$. The two short fibers at the
ends of $\phi^{-t}W^u_{x,R}$, which we left out, can be neglected
since their relative lengths are less than $c_{14}e^{-a_{14}m}$.
The proof of Theorem~\ref{tmmain} is accomplished.
\section{Concluding remarks}
1. The definition of the uniformity of the distribution of unstable
manifolds can be extended to multidimensional Anosov flows
as follows, see \cite{Si92}. Let $x\in M$ and $\Gamma^u_x$
be a (global) unstable manifold through $x$. Consider a sequence
of open subsets $U_j\subset\Gamma^u_x$ of finite diameter (in
the inner metric on $\Gamma_x^u$) such that
(a) $U_1\subset U_2\subset\cdots$ and $\cup_j U_j=\Gamma_x^u$;
(b) for any $R>0$ let $U_j(R)\subset U_j$ be the subset of points
whose distance to $\partial U_j$ is greater than $R$; then
$$
\lim_{j\to\infty}\nu^u_{U_j}(U_j(R)) = 1
$$
where $\nu^u_{U_j}$ is the induced probability measure on
the manifold $U_j$ defined in the same way as in (\ref{nuxR}).
{\bf Definition} \cite{Si92}. The unstable manifold $\Gamma_x^u$ is
said to be uniformly distributed if for any continuous function
$F(x)$ on $M$ we have
$$
\lim_{j\to\infty}\int_{U_j} F\, d\nu^u_{U_j} = \int_M F\, d\mu
$$
for any sequence $U_j\subset \Gamma_x^u$ specified above.
Sinai proved \cite{Si92} that for geodesic flows on compact
manifolds of negative sectional curvature the horocycles
(unstable manifolds) are uniformly distributed.
The extension of Theorem~\ref{tmmain} to multidimensional
Anosov (or geodesic) flows is likely to be true but
not available yet, because
the results of \cite{Ch95} which we used here are completed
only for 3-D flows. \medskip
2. The rate of convergence established by Theorem~\ref{tmmain}
does not seem to be optimal. It is natural to assume that the
square root can be removed in (\ref{main}), see remarks in
the introduction to \cite{Ch95}. Then one gets an algebraic
bound const$\cdot R^{-a}$, $a>0$, which looks more like
an optimal one. \medskip
{\bf Acknowledgements}. I am grateful to Ya.G. Sinai for suggesting
this problem to me and for stimulating discussions. This research was
partially supported by NSF grant DMS-9401417.
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\end{document}
\end